Stochastic Integration with Respect to Fractional Brownian Motion

STOCHASTIC INTEGRATION WITH RESPECT TO FRACTIONAL BROWNIAN MOTION

PHILIPPE CARMONA, LAURE COUTIN, AND GERARD´ MONTSENY

ABSTRACT. For every value of the Hurst index H (0,1) we deﬁne a stochastic integral with respect to fractional Brownian∈ motion of index H. We do so by approximating fractional Brownian motion by semi- martingales. Then, for H > 1/6, we establish an Ito’sˆ change of variables formula, which is more precise than Privault’s Ito formula [24] (established for every H > 0), since it only involves anticipating integrals with respect to a driving Brownian motion.

RESUM´ E´ Pour tout H (0,1), nous construisons une integrale´ stochas- tique par rapport au mouvement∈ Brownien fractionnaire de parametre` de Hurst H. Cette integrale´ est basee´ sur l’approximation du mouvement Brownien fractionnaire par une suite de semi-martingales. Ensuite, pour H > 1/6, nous etablissons´ une formule d’Itoˆ qui precise´ celle obtenue par Privault [24], au sens ou` elle ne comporte que des integrales´ anticipantes par rapport a` un mouvement Brownien directeur.

1. INTRODUCTION Fractional Brownian motion was originally deﬁned and studied by Kol- mogorov [14] within a Hilbert space framework. Fractional Brownian motion of Hurst index H (0,1) is a centered Gaussian process W H with covariance ∈

1 E W HW H = (t2H + s2H t s 2H)(s,t 0) t s 2 − | − | ≥ 1 (for H = 2 we obtain standard Brownian motion). Fractional Brownian motion has stationary increments

E (W H(t) W H(s))2 = t s 2H (s,t 0), − | − | ≥ and is H-self similar 1 ( W H(ct);t 0)=(d W H(t);t 0)(for all c > 0). cH ≥ ≥

Date: December 6, 2001. 1991 Mathematics Subject Classification. Primary 60G15, 60H07, 60H05 ; Secondary 60J65, 60F25 . Key words and phrases. Gaussian processes, Stochastic Integrals, Malliavin Calculus, fractional integration. 1 2 P. CARMONA, L. COUTIN, AND G. MONTSENY The Hurst parameter H accounts not only for the sign of the correlation of the increments, but also for the regularity of the sample paths. Indeed, 1 1 for H > 2 , the increments are positively correlated, and for H < 2 they are negatively correlated. Furthermore, for every β (0,H), its sample paths are almost surely Holder¨ continuous with index ∈β. Finally, it is worthy of 1 note that for H > 2 , according to Beran’s definition [3], it is a long memory 2H 2 process: the covariance of increments at distance u decrease as u − . These significant properties make fractional Brownian motion a natural can- didate as a model of noise in mathematical finance (see Comte and Re- nault [5], Rogers [26]), and in communication networks (see, for instance, Leland, Taqqu and Willinger [16]). Recently, there has been numerous attempts at defining a stochastic integral 1 H with respect to fractional Brownian motion. Indeed, for H = 2 W is not a semi-martingale (see, e.g., example 2 of section 4.9.13 of Liptser6 and Shyr- iaev [19]), and usual Ito’sˆ stochastic calculus may not be applied. However, the integral t H (1.1) Z a(s)dW (s) 0 may be defined for suitable a. On the one hand, since W H has almost its sample paths Holder¨ continuous of index β, for any β < H, the integral (1.1) exists in the Riemann-Stieljes sense (path by path) if almost every sample 1 β path of a has finite p-variation with p + > 1 (see Young [32]): this is the 1 approach used by Dai and Heyde [8] and Lin [18] when H > 2 . Let us recall that the p-variation of a function f over an interval [0,t] is the least upper p bound of sums ∑i f (xi) f (xi 1) over all partitions 0 = x0 < x1 < ... < | − − | xn = T. A recent survey of the important properties of Riemann-Stieltjes integral is the concentrated advanced course of Dudley and Norvaisa [11]. An extension of Riemann-Stieltjes integral has been defined by Zahle¨ [33], see also the recent work of Ruzmaikina [29], by means of composition formu- las, integration by parts formula, Weyl derivative formula concerning fractional integration/differentiation, and the generalized quadratic variation of Russo and Vallois [27, 28]. On the other hand, W H is a Gaussian process, and (1.1) can be defined for deterministic processes a by way of an L2 isometry: see, for example, Nor- ros, Valkeila and Virtamo [21] or Pipiras and Taqqu [23]. With the help of stochastic calculus of variations (see [22]) this integral may be extended to random processes a. In this case, the stochastic integral (1.1) is a di- vergence operator, that is the adjoint of a stochastic gradient operator (see the pioneering paper of Decreusefond and Ustunel [9]). It must be noted that Duncan, Hu and Pasik-Duncan [12] have defined the stochastic integral in a similar way by using Wick product. Feyel and de La Pradelle [13], Ciesielski, Kerkyacharian and Roynette [4] also used the Gaussian property of W H to prove that W H belongs to suitable function spaces and construct a stochastic integral. 3 Eventually, Alos, Mazet and Nualart [1] have established, following the ideas introduced in a previous version of this paper, very sharp sufficient conditions that ensures existence of the stochastic integral (1.1). The construction of the stochastic integral. The starting point of our approach is the following representation of fractional Brownian motion given by Decreusefond and Ustunel [9]

t H H W (t) = Z K (t,s)dB(s), 0 where H 1 (t s) 2 1 1 1 t (1.2) KH(t,s) = − − F(H , H,H + ,1 )(s < t), Γ 1 2 2 2 s (H + 2 ) − − − where F denotes Gauss hypergeometric function and (B(t);t 0) is a standard Brownian motion. ≥ The first step we take is to define the integral of deterministic functions H H H (section 4). Observing that K (t,s) = It 1[0,t](s) where It is the integral operator t H H ∂ H (1.3) It f (s) = K (t,s) f (s) + Z ( f (u) f (s)) 1K (u,s)du, s − we define t t H H (1.4) Z f (s)dW (s) = Z It f (u)dBu 0 0 for suitable deterministic functions. In order to extend formula (1.4) to random processes f , we introduce the key idea of this paper, which is to approximate W H by processes of the type t (1.5) WK(t) = Z K(t,s)dB(s)(t [0,T]) 0 ∈ with a kernel K smooth enough to ensure that WK is a semi-martingale (Proposition 2.5). Then, for a such a semi-martingale kernel K and a good integrand a, we prove the following generalization of (1.4)

t t t (1.6) Z a(s)dWK(s) = Z Ita(s)dB(s) + Z ItDsa(s)ds, 0 0 0 H where Dsa denotes the stochastic gradient and It is defined as It . In sections 2 and 3 topologies are defined on the space of kernels and the space of integrands. These are not ’natural’ topologies but ad hoc ones, whose sole aim is to fulfill the following requirements: 1. The mapping (a,K) ta(s)dW (s) is bilinear continuous (Proposi- R0 K tions 6.1). → 2. Rough kernels such as KH are in the closure of the space of semi- martingale kernels. 4 P. CARMONA, L. COUTIN, AND G. MONTSENY 3. For nice integrands a the process t ta(s)dW (s) has a continuous R0 K modification (Theorem 7.1). → Eventually, we are able to take limits in the Ito’sˆ formula established for n semi-martingale kernels in Proposition 8.1. Let Cb be the set of functions f whose derivatives, up to the order n N, are continuous and bounded. Privault proved that for f C2, ∈ ∈ b t t H H H 1 H 2H 1 f (W (t)) = f (0)+Z f 0(W (s)).dW (s)+ Z f 00(W (s))(cH2Hs − )ds, 0 2 0 t H H 2 where 0 f 0(W (s)).dW (s) is the L -limit of divergences (Skorokhod in- R 3 tegrals). For H > 1/4 and f Cb, we show that this integral, which is not in general the limit of Riemann∈ sums, is a Skorokhod integral with respect to the driving Brownian motion: t H H H (1.7) f (W (t)) = f (0) + Z It f 0(W )(s)dBs 0 t 1 H 2H 1 + Z f 00(W (s))(cH2Hs − )ds. 2 0 By the same procedure, we have been able to prove an Ito’sˆ formula for H > 1/6, in Proposition 8.11. We do not give here this formula since it is complex (5 lines) and does not seem to be an easy starting point for a generalization to every H (0,1). ∈ To end this rather lengthy introduction we try to give an answer to a question that nearly everyone involved with stochastic integration with respect to Fractional Brownian motion has asked us. “What is the difference between the stochastic approach and the pathwise approach to integration with respect to Fractional Brownian motion?” H H On the one hand we prove in the Appendix that for W1 ,W2 two independent fractional Brownian motions of index 1/4 < H < 1/2, the integral t H H Z W1 (s)dW2 (s) 0 cannot be defined for the classical and generalized pathwise integrals. On the other hand, a slight generalization of our results enabled Coutin and Qian [7, 6] to show that for 1 < H < 1 the integral tW H(s)dW H(s) can be 4 2 R0 1 2 defined as a process with a continuous modification. Acknowledgement We want to thank an anonymous referee, not only for correcting some mistakes, but also for detailed and useful comments. Our main source of inspiration for Ito’sˆ formula is T. Lyons paper [20], and in particular the idea that in order to integrate with respect to rough signals, you may sometimes have to replace approximating Riemann sums, of order 1, with Taylor sums of higher order. More precisely, you may want H H to replace f (W (ti+1)) f (W (ti)) by − H H H H H f (W (ti+1)) f (W (ti)) W (ti+1) W (ti) f 0(W (ti)) + ... − − − 5

2. THEVECTORSPACESOFKERNELS The spaces of integrands and kernels we shall work with depend on parameters (p,γ,θ). We shall assume from now on that the set (p,γ,θ) of parameters is admissible, that is

1 1 2 2p (2.1) 1 < p < 2, + = 1, > γ > 1/q,δ = p q q 2 p − 1 2 γ 1 1 (2.2) hγ = θ <= , θ + θ = 1. ∗ q − /2 ∗/2 δ 2 Observe that /2 and q/2 are conjugated Holder¨ exponents, and that 1 > q . Let K : R2 R be a measurable kernel such that → t 2 Z K(t,s) ds < +∞ for every t > 0(2.3) 0 (2.4) K(t,s) = 0 if s > t . We consider the adapted Gaussian process t WK(t) = Z K(t,s)dBs , 0 where (Bt ;t 0) is a standard Brownian motion. ≥ Definition 2.1. We say that K is a rough kernel if there exists a measurable function (t,s) ∂1K(t,s), such that u ∂1K(u,s) is integrable on every [t,t ] (s,+∞)→and satisfies → 0 ⊂ t0 (2.5) K(t0,s) K(t,s) = Z ∂1K(u,s)du (s < t < t0). − t We say that K is a smooth kernel if K is a rough kernel that satisfies t (2.6) K(t,s) K(s,s) = Z ∂1K(u,s)du (s < t). − s The topology on the space of rough kernels is defined via mixed Lebesgue spaces (see Stroock [31], section 6.2). Let (E1,B1,µ1) and (E2,B2,µ2) be a pair of σ-finite measure spaces and let p1, p2 [1,∞). Given a measurable ∈ function f on (E1 E2,B1 B2), define × × 1 p2/p1 p2 f = " f (x ,x ) p1 µ (dx ) µ (dx )# , p1,p2 Z Z 1 2 1 1 2 2 k k E2 E1 | | (p ,p ) and let L 1 2 (µ1,µ2) denote the mixed Lebesgue space of R valued, B1 × B2 measurable f for which f < ∞. k kp1,p2 For i = 1,2 we let (Ei,Bi,µi) = ((0,t),B(0,t),ds) be the space (0,t) endowed with the σ-field of Borel sets and Lebesgue measure, p1 = p and δ ∂ γ def p2 = , f (u,s) = 1K(u,s)(u s) 1(0

1/δ t t δ/p def ∂ γ p ! (2.7) K γ,θ,p,t = Z Z 1K(u,s)(u s) du ds k k 0 s | − |

1 t θ θ + Z K(t,s) ds . 0 | |

Let Eγ,θ,p,t be the space of rough kernels K such that K γ θ < +∞. k k , ,p,t Lemma 2.2. Eγ,θ,p,t is a Banach space.

Proof. Assume that (Kn)n N is a Cauchy sequence in Eγ,θ,p,t. Then the ∈ γ sequence of functions fn(u,s) = ∂1Kn(u,s)(u s) 1(0

φ(s) t z(u,s)du if 0 < s < v < t K(v,s) = ( − Rv 0 if v s t ≤ ≤ we conclude that K is a rough kernel for which K Kn γ,θ,p,t 0 since by construction : k − k →

K Kn γ θ = f fn δ + Kn(t,.) φ(.) θ . k − k , ,p,t k − kp, k − k

Remark 2.3. It is quite straightforward to prove, via Holder’s¨ inequality, that for t T, the space Eγ,θ,p,T can be continuously embedded in Eγ,θ,p,t. ≤ ∞ Indeed, we only need to prove that K(T, ) K(t, ) Lθ(0,t) < + . We ﬁrst observe that fors < t: k · − · k T T 1/p 1 γ γ p K(T,s) K(t,s) = Z ∂1K(u,s)du (t s) q − Z ∂1K(u,s)(u s) du . | − | t ≤ − t | − |

Therefore, applying Holder’s¨ inequality to the pair of conjugated exponents (r = δ/θ,r0) 1/r t θ t θ 1 γ 0 θ ( q )r0 Z K(T,s) K(t,s) ds C Z (t s) − ds K γ,θ,p,T . 0 | − | ≤ 0 − k k 2 1 The ﬁrst factor may be majorized by a constant C, since γ > θ implies q − ∗ θ( 1 γ)r > 1. q − 0 Let us recall that our main results are proved by approximating the fractional Brownian motion kernel by smooth kernels for which WK is a semi- martingale. 7 Deﬁnition 2.4. We say that the smooth kernel K is a semi-martingale kernel if u t 2 2 (2.8) supZ ∂1K(u,s) ds + Z K(s,s) ds < +∞ ( t > 0). u t 0 0 ∀ ≤

Proposition 2.5. (1) If K is a semi-martingale kernel, then WK is a semi- martingale with decomposition t t

WK(t) = Z K(s,s)dBs + Z W∂1K(s)ds. 0 0 (2) The vector space Fγ,θ,p,t of semi-martingale kernels in Eγ,θ,p,t is dense in Eγ,θ,p,t.

Proof. (1) The deﬁnition of a smooth kernel is more than what we need to apply Fubini’s stochastic Theorem (see, e.g., Protter [25] Theorem 46): t t WK(t) = Z (K(s,s) + Z ∂1K(u,s)du)dBs 0 s t t u = Z K(s,s)dBs + Z (Z ∂1K(u,s)dBs)du. 0 0 0 (2) Let ε > 0 be given. We perform the change of variables u = v + s in the ∂ γ deﬁnition of K γ,θ,p,t, and we let g(v,s) = 1K(v + s,s)v 1(0 2 . Hence for ε > 0 we consider

t γ (2.9) Λε(r,s) = K(t,s) Z h(u s,s)(u + ε s)− du (s < r < t) − r − − The Λε is a semi-martingale kernel, and δ t t γ /p Λ Λ δ u s ε γ,θ,p,t (const)Z dsZ − 1 du 0 k − k ≤ 0 s u + ε s − → − as ε 0 by dominated convergence. → 8 P. CARMONA, L. COUTIN, AND G. MONTSENY Deﬁnition 2.6. In order to prove the continuity of our stochastic integral β we introduce Aγ,θ,p,T the subset of Eγ,θ,p,T of kernels K such that

t t+τ 2 2 (hγ+β) β K β = sup τ− Z Z (u s) ∂1K(u,s)du ds < +∞ k k 0

K = K γ θ + K β . k| |k k k , ,p,T k k Since Ito’sˆ formula is proved by approximating with smooth kernels, we β β introduce Bγ,θ,p,T the closure of Fγ,θ,p,T in Aγ,θ,p,T . 2.1. The case of fractional Brownian motion. We recall here the basic properties of hypergeometric functions required throughout this paper (see e.g. Chapter 9 of Lebedev [15]). Gauss hypergeometric function F(α,β,γ,z) is deﬁned for every α,β, γ, z < 1 and γ = 0, 1,... by | | 6 − α β α β γ ( )k( )k k F( , , ,z) = ∑ γ z , k 0 ( )kk! ≥ Γ(α + k) where (α)0 = 1 and (α)k = = α(α+1)...(α+k 1) is the Pocham- Γ(α) − mer index. The convergence radius of this series is 1 and, as soon as Re(γ β α) > 0, limz 1 F(α,β,γ,z) exists and is ﬁnite. Obviously we have − − →

F(α,β,γ,z) = F(β,α,γ,z), F(α,β,γ,0) = 1. Furthermore, if arg(1 z) < π and Re(γ) > Re(β) > 0, | − | Γ γ 1 ( ) β 1 γ β 1 α (2.10) F(α,β,γ,z) = Z u − (1 u) − − (1 zu)− du. Γ(β)Γ(γ β) 0 − − − The hypergeometric function F(α0,β0,γ0,.) is said to be contiguous to F(α,β,γ,.) if α α0 = 1 or β β0 = 1 or γ γ0 = 1. If F1 and F2 are hypergeometric functions| − | contiguous| − to|F, then| there− exists| a relation of the type

P(z)F(z) + P1(z)F1(z) + P2(z)F2(z) = 0 ( arg(1 z) < π), | − | with P(z),P1(z),P2(z) polynomials in the variable z. These relations ensure that there exists an analytical continuation of F(α,β,γ,z) to the domain

C C (C 0, 1, 2,... ) z : arg(1 z) < π . × × \{ − − } × { | − | } We shall also use the asymptotic estimate, for arg(1 z) < π,Re(Γ) > Re(α) > 0,Re(β) < 0: | − |

Γ(γ)Γ(α β) β F(α,β,γ,z) − ( z)− ( z ∞). ∼ Γ(α)Γ(γ β) − | | → − 9

1 H Lemma 2.7. Given s > 0 and H = 2 , the function t K (t,s) is differentiable on (s,+∞) with derivative 6 → 1 H H (s/t) 2 − H 3/2 ∂1K (t,s) = (t s) − (0 < s < t). Γ(H 1 ) − − 2 H β Furthermore, K Aγ θ for admissible parameters (p,q,γ,...) such that ∈ , ,p,T 1 1 + γ > 3/2 H , θ H + 1 > 0 and β + H < 1. p − − − 2

ε 1 In particular, for 0 < < H < 2 , there exists a set of admissible parameters for which hγ = H ε. − Remark 2.8. Observe that for H < 1 , we have γ 1 > 1/2 H > 0. 2 − q − Proof. Let us state three facts: (1) From the analyticity of the Gauss function, we deduce that the function t KH(t,s) is differentiable on (s,+∞) with a derivative f (H,t,s) such → C 1 that H f (H,t,s) is holomorphic on H : Re(H) > 2 . (2) The→ integral representation (2.10) implies ∈ that for H (−1/2 ,1) the func- H ∈ H tion t K (t,s) is differentiable on (s,+∞) with derivative ∂1K (t,s). → H (3) The function H ∂1K (t,s) is holomorphic in the region → 1 ∆ = H C : 0 < Re(H) < 1 . { ∈ }\ 2 H Therefore, by analytic continuation: for 0 < s < t, ∂1K (t,s) = f (H,t,s) on ∆. H To determine the admissible set of parameters for which K Eγ,θ,p,T , we ∈ just use the fact that (s,t) KH(t,s) is continuous on 0 < s < t , com- bined with the following consequence→ of the asymptotic estimate{ of}F, H H 1 K (t,s) s 0+ (const)s− − 2 ∼ → | | H H 1 K (t,s) s t (const)(t s) − 2 ∼ → − − 1 1 1 ε 1 Therefore, given 0 < ε < H < , we can set = and hγ = θ = H ε, 2 q 2 − 3 ∗ − 2 γ = H ε/2 to get q > 2, 2/q > γ, and γ 1 = 1 (H ε/2) > 1 H > q − − − q q − − 2 − 0. It remains to show that K β < +∞ (indeed, we can then approximate K k k 1 by the sequence Kn(t,s) = K(t + n ,s) of smooth kernels in Fγ,θ,p,T ) : for 0 < t < t + τ T, ≤ t t+τ 2 β Z Z (u s) ∂1K(u,s)du ds 0 t − ≤ t t+τ 2 1+2ε β+H 1 ε Z (t s)− Z (u s) − − du ds 0 − t − 2(β+H ε) Cτ − ≤ 10 P. CARMONA, L. COUTIN, AND G. MONTSENY

2.2. Properties of Integral Operators associated to Kernels. Let X be a separable Hilbert space with norm X , and K be a rough kernel. To a measurable a : [0,T] X we associate|·| → t K ∂ (2.11) It (a)(s) = K(t,s)a(s) + Z (a(u) a(s)) 1K(u,s)du, 0 − as soon as the integral in the right hand side makes sense for almost every s [0,T]. ∈ K Similarly we shall denote by Jt the integral operator deﬁned on measurable f : [0,T]2 X by → t K ∂ (2.12) Jt ( f )(s) = Z f (u,s) 1K(u,s)du, s as soon as the integral in the right hand side makes sense for almost every s [0,T]. Observe∈ that if ∆0 a = a(u) a(s), then s,u − K K ∆0 It (a)(s) = K(t,s)a(s) + Jt ( a)(s).

Since K and ∂1K need not be positive, we introduce a dominating positive kernel

T K+(t,s) = K(T,s) + Z ∂1K(u,s) du1(s

(2.13) K(t + τ,s) K(t,s) K+(t,s) K+(t + τ,s) | − | ≤ − K(t,s) K+(t,s). | | ≤

For β R,β = 0, we let wβ be the weight ∈ 6 β wβ(u,s) = max(u s,0) . − Then if duds denotes the Lebesgue measure on [0,T]2, we introduce the Lebesgue space

q q 2 Lwβ (X) = L ([0,T] ;X;wβ(u,s)duds). Eventually we denote by H β(X) the set of β Holder¨ continuous functions taking their values in X: 11

f (u) f (s) β ( X ∞) H (X) = f : [0,T] X, f H β(X) = sup | − β | < + → k k 0 u

˜ β 2 f (u,s) X (X) = ( f : [0,T] X, f β = sup < +∞) H H˜ (X) | β| → k k 0 u ~~0 and β R∗. The constant C may vary from line to line but depends only on (γ,∈δ, p,T,β) ; to ease notation, we shall omit this depen- dency when the context allows it.~~

Proposition 2.9. For every ﬁxed t [0,T] ∈ K + K+ 1. The mappings J : ( f ,K) J ( f )(t) and J : ( f ,K) Jt ( f ) are →q 2 →2 bilinear continuous from Lw γq (X) Eγ,θ,p,T to L ([0,T] ;ds,X). − ×K 2. The mapping It : (a,K) It (a) is bilinear continuous θ q ∗ → ∆0 from L ([0,T];ds;X) a : a Lw γq (X) Eγ,θ,p,T to ∩ ∈ − × L2([0,T]2;ds,X).

Proof. Since JK( f ) JK+ ( f ), we can suppose, without any loss in t X ≤ t | |X generality, that X = R and study JK+ . First, apply Holder’s¨ inequality to the pair of conjugated exponents (p,q):

~~1 1 t q t p K+ q ∂ p Jt ( f )(s) Z w γq(u,s) f (u,s)du Z wγp(u,s) 1K(u,s) du . ≤ s − | | × s | |~~

~~Then, apply Holder’s¨ inequality to (q/2,δ/2) to get~~

~~K+ Jt ( f )(.) f Lq (R) K γ,θ,p,T . L2([0,T],ds) w γq ≤ k k − k k~~

~~The second assertion is clearly a consequence of the ﬁrst one, another Holder’s¨ inequality for (θ/2,θ∗/2), and Remark 2.3:~~

~~K(t, )a( ) 2 a θ K γ θ C a θ K γ θ . k · · kL ([0,T],ds) ≤ k kL ∗ ([0,T];R) k k , ,p,t ≤ k kL ∗ ([0,T];R) k k , ,p,T~~

~~The next proposition establish the Holder¨ regularity of a rough Kernel K(t,s) with respect to the ﬁrst variable t. 12 P. CARMONA, L. COUTIN, AND G. MONTSENY~~

~~Proposition 2.10. The space Eγ,θ,p,T may be continuously embedded into H hγ (L2([0,T];ds)). Furthermore, the mapping K K+ is continuous from Eγ,θ,p,T into → H hγ (L2([0,T];ds)).~~

Proof. According to the inequalities (2.13), we only need to prove the second point. The embedding is the following : we identify K+ with the 2 function F : [0,T] X = L ([0,T];ds) deﬁned by F(t) = K+(t, ). Let 0 t t + τ T ;→ we have, for s = t: · ≤ ≤ ≤ 6 K+ K+(t + τ,s) K+(t,s) = K(T,s) 1 τ (s) + J ( f + g)(s) − | | (t,t+ ) T with f (u,s) = 1(t,t+τ)(s)1(t+τ,T)(u), g(u,s) = 1(0,t)(s)1(t,t+τ)(u) According to Lemma B.1, there exists a constant C such that: 2/q γ f q + g q C τ − Lw γq Lw γq k k − k k − ≤ | | and θ τ1/ ∗ 1(t,t+τ) θ C . L ∗ ([0,T],ds) ≤ 1 Therefore, since hγ = θ < 2/q γ, we only need to apply Proposition 2.9 ∗ − to f ,g,1 τ to obtain the continuity of K K+. (t,t+ ) →

~~Similarly, we can establish Holder¨ regularity for the operators J and J+.~~

~~Proposition 2.11. The operators J and J+ are bilinear continuous from ˜ β β β+h 2 H (X) Aγ θ to H (L ([0,T]);ds;X) as soon as 0 < h < hγ. × , ,p,T~~

Proof. Let 0 t t + τ T and K˜ be one of K,K+. Then, ≤ ≤ ≤ 0 K˜ K˜ K˜ K˜ (2.14) ∆ τJ (a)(s) = J τ(a)(s) J (a)(s) = J τ(a)(max(s,t)). t,t+ t+ − t t+ We plug in the estimate a(u,s) a ˜ β wβ(u,s) for u > s, and get | | ≤ k kH ∆0 K˜ ∆0 K˜ t,t+τJ (a)(s) t,t+τJ (wβ)(s) a ˜ β ≤ k kH and thus the Proposition is an easy consequence of the following Lemma.

K˜ β Lemma 2.12. The mapping K J (wβ) is linear continuous from Aγ,θ,p,T β → to H +h(L2([0,T]);ds) as soon as 0 < h < hγ. 13 Proof. For 0 < t < t + τ T ≤ K+ K+ K+ K+ J τ(wβ)(s) J (wβ)(s) = J τ(1 τ (s)wβ)(s) + J τ(1 (s)wβ)(s). t+ − t t+ (t,t+ ) t+ (0,t) q According to Lemma B.1, the norm of (u,s) 1(t,t+τ)(s)wβ(u,s) in Lw γq → − is dominated by Cτβ+hγ . Therefore, Proposition 2.9

K+ β+hγ J τ(1 τ (s)wβ)(s) Cτ K γ θ . t+ (t,t+ ) 2 , ,p,T L ([0,T],ds) ≤ k k β Since K+ Aγ θ , then for h < hγ, ∈ , ,p,T K+ β+h J τ(1 (s)wβ)(s) Cτ K β t+ (0,t) 2 L ([0,T],ds) ≤ k k

~~Combining these two upper bounds yields the desired result.~~

~~The following technical results now involve pairs of rough kernels.~~

Lemma 2.13. The mapping (K1,K2) K1(t, ),K2(t, ) L2([0,T]) is bilinear 2 → h · · i continuous from Eγ,θ,p,T to the set of absolutely continuous functions, van- ishing at the origin, endowed with the norm T f = Z f˙ (s)ds k k 0 where f˙ is equal almost everywhere to the derivative of f . Proof. Assume that the kernels can be written, as in the proof of Proposi- tion 2.5, (2.15) T Ki(t,s) = Ki(T,s) Z hi(u s,s)w γ(u + ε,s)du for s t, i = 1,2 − t − − ≤ ∑n φ ψ φ ψ ∞ where hi(v,s) = l=1 l,i(v) l,i(s) and the l,i, l,i L (0,T) have disjoint supports. Then the mapping ∈

~~Λ : t K1(t, ),K2(t, ) 2 → h · · iL ([0,T]) is differentiable with derivative~~

~~t Λ˙ (t) = K1(t,t)K2(t,t) + Z (∂1K1(t,s)K2(t,s) + K1(t,s)∂1K2(t,s))ds. 0 From the domination relation (2.13) we get ˙ d (2.16) Λ(t) K1,+(t, ),K2,+(t, ) 2 ≤ dt h · · iL ([0,T])~~

~~Integrating with respect to t yields, taking into account Proposition 2.10,~~

~~T Λ˙ Z (t) dt K1,+(t, ),K2,+(t, ) L2([0,T]) C K1 γ,θ,p,T K2 γ,θ,p,T 0 ≤ h · · i ≤ k k k k~~

14 P. CARMONA, L. COUTIN, AND G. MONTSENY By Proposition 2.5, the space of kernels that can be written as in (2.15) is dense in Eγ,θ,p,T . This concludes our proof since the space of absolutely continuous functions is closed.

~~We now associate to a pair (K,K0) of rough kernels an integral operator t K,K0 ∂ (2.17) J2,t (a)(s,s1) = a(s,s1)Z K(u,s) 1K0(u,s1()) du1(s1~~

~~Lemma 2.14. If 0 < h < hγ, then the mapping J2,t is bilinear continuous β ˜ β +hγ β+2h 2 2 from H (X) Eγ,θ,p,T Aγ θ into H (L ([0,T] ;X;dsds1)). × × , ,p,T~~

~~Proof. According to equality (2.14) applied to~~

~~a˜(u,s1) = K(u,s)a(s,s1)1(s1~~

~~0 K,K0 K+0 ∆ J (a)(s,s ) a β J τ(K (u,s)wβ(u,s ))(s )1 t,t+τ 2 1 ˜ t+ + 1 1 (s>s1) ≤ k kH According to Proposition 2.10 there exists a constant C such that:~~

~~(2.18) (u,s) K+(u,s)wβ(u,s1) ˜ β+hγ C K γ θ . → H (L2([0,T])) ≤ k k , ,p,T~~

~~Therefore, thanks to Proposition 2.11 we have the upper bound~~

~~0 K,K0 β+2hγ ∆ τJ (a)(s,s1) C K γ θ K0 β+hγ a ˜ β τ . t,t+ 2 2 , ,p,T A H L ([0,T],X,ds) ≤ k k γ,θ,p,T k k~~

Lemma 2.15. The mapping (K,K ) 0 → K(t + τ, ) K(t, ), K (t, ) 2 is bilinear continuous from h · − · 0 · iL ([0,T]) β 2 β+h Aγ θ Eγ θ to R with norm bounded by Cτ for any h (0,hγ) (we , ,p,T × , ,p,T ∈ extend K so that K(u,s) = 0 if u < s). 15

~~Proof. Thanks to (2.13) we can restrict ourselves to K+,K+0 . we have:~~

( + τ, ) ( , ), ( , ) K+ t K+ t K+0 t L2([0,T]) · − · · t t+τ ∂ = Z dsK+0 (t,s)Z du 1K+(u,s) 0 t t t+τ β β∂ = Z dsK+0 (t,s)Z du(u s)− (u s) 1K+(u,s) 0 t − − t t+τ β β∂ Z dsK+0 (t,s)(t s)− Z du(u s) 1K+(u,s) ≤ 0 − t − t β ∆0 K+ = Z dsK+0 (t,s)(t s)− t,t+τJ (wβ)(s) 0 − 1 1 t t β 2 2 2 2 2 ∆0 K+ Z dsK+0 (t,s) (t s)− Z ds t,t+τJ (wβ)(s) ≤ 0 − 0 τβ+h (C1 K0 β ) (C2 K γ,θ,p,T ). ≤ Aγ,θ,p,T × k k

The second inequality is Cauchy-Schwarz’s inequality, and the last one is due to Holder’s¨ inequality (θ/2,θ∗/2) for the ﬁrst factor (since βθ∗ < 1) and Lemma 2.12 for the second factor. We can now conclude with the help of Remark 2.3.

3. THE SPACE OF INTEGRANDS 3.1. A review of basic notions of Malliavin Calculus. A nice introduction to Malliavin Calculus can be found in Nualart’s book [22], but for the sake of completeness, we state here the few definitions and properties we use in this paper. Let Ω denote the space C(I,R), I = [0,T], equipped with the topology of uniform convergence on the compact sets, F the Borel σ field on Ω, P − the standard Wiener measure, and let Bt(ω) = ω(t), 0 t T . For any { ≤ ≤ } t 0, we define Ft = σ(ω(s), s t) N , where N denotes the class of the elements≥ in F which have zero ≤P measure.∨ For h L2(I,R), we denote by B(h) the Wiener integral ∈

~~B(h) = Z (h(t),dBt). I~~

~~2 i Let X be a separable Hilbert space with norm X . (Usually X = L ([0,T] ;R)). Let S denote the dense subset of L2(Ω,F ,P)|·consisting| of those classes of random variables of the form :~~

(3.1) F = f (B(h1),...,B(hn)), 16 P. CARMONA, L. COUTIN, AND G. MONTSENY N ∞ Rn 2 R where n , f Cb ( ;X), h1,..,hn L (I, ). If F has the form (3.1), ∈ ∈ ∈ def we deﬁne its derivative as the process DF = DtF, t I given by { ∈ } n ∂ f DtF = ∑ ∂ (B(h1),...,B(hn))hk(t). k=1 xk We shall denote by D1,α the closure of S with respect to the norm

F 1,α = F α + DF 2 α. k k k k kk kL (I)k 1/α t α/2 α 1/α 2 = E F + E"Z DuF du # | | 0 | | The higher order derivative DnF are deﬁned inductively, and the space Dn,α is the closure of S under the norm n = + i . F n,α F α ∑ D F L2(Ii) k k k k i=1 α Then, deﬁne δ, the Skorokhod integral with respect to W, as the adjoint of D, i.e. , Dom(δ) is the set of u L2(Ω I) such that there exists a constant c with ∈ ×

~~EZ DtFutdt c F 2, F S. I ≤ k k ∀ ∈~~

~~If u Domδ, δ(u) is deﬁned as the unique element of L2(Ω) which satisﬁes ∈~~

E[δ(u)F] = EZ DtFutdt, F S. I ∀ ∈ In order to prevent a confusion between δ, an admissible parameter, and δ the Skorokhod integral, we shall from now on use the same notations for the Ito and the Skorokhod integral, that is: T δ(u) = Z u(s)dBs . 0 Let L1,2 be the Hilbert space L2(I;D1,2) endowed with the norm

~~2 E 2 2 u L1,2 = Z ut dt + Z (Dvut) dvdt. k k I I I × We have L1,2 Domδ, and for u L1,2 ⊂ ∈ (3.2) T 2 E" # E 2 E 2 Z u(s)dBs = Z ut dt + Z Z Dtus Dsut dsdt u L1,2 . 0 I I I ≤ k k~~

2 Note that u L (Ω I); u is Ft progressively measurable Domδ, and for such a{u, ∈δ(u) coincide× with the usual Itoˆ integral. Note} that ⊂ when u is progressively measurable, Dsut = 0 for s > t, so (3.2) is consistent with the formula in the adapted case. 17 Let L2,2 be the Hilbert space L2(I2,D2,2) endowed with the norm (3.3) 2 E 2 2 2 2 u L2,2 = Z u (t,s)dtds + Z (Dvu(t,s)) dvdtds + Z (Dv,ru(t,s)) dtdsdvdr. k k I2 I3 I4 We have L2,2 Dom(δ δ) and for u L2,2 ⊂ ◦ ∈ T s E 2 (3.4) Z Z u(s,s1)dBs1 dBs 2 u L2,2 . 0 0 ≤ k k

~~Property P : Integration by parts formula Suppose that u belong to L1,2. Let F be a random variable belonging to D1,2 such that E F2 u2dt < ∞, RI t then~~

~~(3.5) Z FutdBt = F Z utdBt Z DtFutdt, I I − I in the sense that Fu belong to Dom(δ) if and only if the right hand side of (3.5) belongs to L2(Ω).~~

For m = 1,2, let C βLm,2(0,t) = H β(Dm,2) Lm,2 be the space of processes a Lm,2(0,t) such that for a constant C ∩ ∈ β (3.6) a(u) a(v) m,2 C u v (0 u,v t). k − kD ≤ | − | ≤ ≤ 3.2. The space of good integrands. Recall that (p,γ,θ) is a set of admissible parameters, that is

1 1 2 2p (3.7) 1 < p < 2, + = 1, > γ > 1/q,δ = p q q 2 p − 1 2 γ 1 1 (3.8) hγ = θ < , θ + θ = 1. ∗ q − /2 ∗/2 1,2 The space GIγ,θ,q,t is the space of adapted processes a L (0,t) such that ∈ a γ θ < ∞ where k k , ,q,t

~~1/q def γ q a γ,θ,q,t = Z ( a(u) a(s) D1,2 (u s)− ) duds k k 0~~

(we hope that the use of the same notations form norms of kernels and norms of integrands will not confuse the reader). With the notations introduced in the previous section, 0 a = ∆ a q + a θ 2 . γ,θ,q,t L (D1,2) L ∗ (L (Ω)) k k w γq k k − 18 P. CARMONA, L. COUTIN, AND G. MONTSENY

Observe that if a CβL1,2(0,t) is adapted and bounded, and if β > γ 1 , ∈ − q then a GIγ,θ,q,t. ∈ Finally, we need to introduce the subset CGIγ,θ,q,t of processes a GIγ,θ,q,t such that ∈ t t 2 q/2 γq Z Z E (Dsa(u)) (u s)− duds < +∞ 0 s − which we endow with the norm

a γ θ = a γ θ + Da q 2 Ω . , ,q,t , ,q,t Lw γq (L ( )) |k k| k k k k − 4. INTEGRATIONOFDETERMINISTICFUNCTIONS We recall that to a rough kernel K we associate the integral operator deﬁned for a measurable a on (0,t)

t (4.1) Ita(s) = K(t,s)a(s) + Z (a(u) a(s))∂1K(u,s)du (0 < s < t) s − as soon as the integral on the right hand side makes sense for almost every s in (0,t). When K is a smooth kernel, we have t Ita(s) = K(s,s)a(s) + Z a(u)∂1K(u,s)du (0 < s < t). s We shall also consider the integral operator

~~t (4.2) Jta(s) = Z a(u,s)∂1K(u,s)du (0 < s < t). s~~

~~We end this section by the deﬁnition of the integral of a deterministic function. Since K(t,s) = It1[0,t](s), we have, for s < t,~~

~~t s Cov(WK(s),WK(t)) = EZ K(t,u)dBu Z K(s,v)dBv 0 0~~

= It1 , It1 [0,t] [0,s] L2 2 where , L2 denotes the inner product of L (R+). h i 2 1 2 Accordingly, we deﬁne LK(0,t) = It− (L (0,t)) and endow it with the inner product def (4.3) f ,g 2 ( , ) = It f , Itg L2 . h iLK 0 t h i 2 2 L (0,t) is the closure of L (0,t) with respect to , 2 . Given f K K LK(0,t) 2 h· ·i ∈ LK(0,t), it is natural to deﬁne t t (4.4) Z f (s)dWK(s) = Z It f (u)dBu. 0 0 19

~~2 This is clearly an isometry between LK(0,t) and the Gaussian space gener- ated by (WK(s),0 s t). It is interesting to note that for f 1 we obtain ≤ ≤ ≡ t Z dWK(s) = WK(t). 0~~

~~5. REGULARITY OF SAMPLE PATHS~~

Proposition 5.1. Assume that K Eγ,θ,p,t. Then, almost surely, the sample ∈ paths of WK are Holder¨ continuous of index τ (0,hγ). More precisely, if 1 1 ∈ α > and τ [0,hγ [ then for some constant C hγ ∈ − α WK(v) WK(u) sup | − τ | C K γ,θ,p,t . 0 u

~~Proof. Since WK is a Gaussian process with covariance min(u,v) (5.1) RK(u,v) = Z K(v,s)K(u,s)ds 0 Proposition 2.10 yields for 0 u < v < t ≤~~

hγ WK(v) WK(u) = K(v, ) K(u, ) 2 C v u K γ θ . k − k2 k · − · kL (0,t) ≤ | − | k k , ,p,t Then, we use Kolmogorov’s continuity Lemma in the special case of Gauss- ian processes, as described by the next Lemma. Lemma 5.2. Let (Z(t),0 t T) be a centered Gaussian process such that for some δ > 0 ≤ ≤ δ E (Z(t) Z(s))2 C(T) t s (0 s,t T). − ≤ | − | ≤ ≤ Then Z has a modiﬁcation (denoted by the same letter Z) with Holder¨ continuous paths of order λ for each λ [0,δ/2[. Furthermore, for every p sup(1,2∈/δ) we have ≥ 1+1/p 1 2 δ/2 sup Z(t) Z(s) C(T) 2 T mp 1 δ/2 0 s,t T | − | ≤ 1 2 p ≤ ≤ p − −

Z(t) Z(s) 1 λ δ 2 λ δ sup | − λ | C0( , , p,T)C(T) (0 < /2 1/p). 0 s 0 and some p 1 ≤ ≤ ≥ p 1+σ E[ Xt Xs ] C t s (0 s,t T). | − | ≤ | − | ≤ ≤ Then for every λ [0,σ/p[ the random variable ∈ X X ( t s ) Mλ = sup | − λ| : s,t Q,s = t,0 s,t T t s ∈ 6 ≤ ≤ | − | is almost surely ﬁnite. More precisely,

λ+1+1/p σ λ 1/p 2 1+ Mλ C T p− . k kp ≤ 1 2λ σ/p − − Proof of Lemma 5.2. For every p 1, since Z(t) Z(s) is a centered Gauss- ian random variable, we have ≥ − δ E[ Z(t) Z(s) p] mpC(T)p/2 t s p/2 (0 s,t,T). | − | ≤ p | − | ≤ If p > sup(1,2/δ), then we obtain the upper bound on the p-th moment, and the fact that Z has Holder¨ continuous paths for every λ [0,δ/2 1/p[. ∈ −

Corollary 5.4. Assume that for some p 1,K Eγ,θ,p,t. Then there exists ≥ ∈ a sequence of kernels (Kn,n N) in Fγ,θ,p,t such that ∈ 2 sup WK(s) WKn (s) 0 in L and a.s. s t | − | → ≤ Proof. There exists a sequence of kernels (Kn,n N) in Fγ,θ,p,t such that ∈ Kn K γ θ 0. We let k − k , ,p,t →

~~Mn = sup WK(s) WKn (s) . s t | − | ≤ α 1 τ Proposition 5.1 yields, for > hγ and = 0,~~

Mn α C K Kn γ θ 0. k k ≤ k − k , ,p,t → α By taking a subsequence, we can ensure that ∑ Mn α < +∞, and thus ob- k k tain, by a Borel-Cantelli argument, the almost sure convergence of Mn to 0.

6. CONSTRUCTIONOFTHE STOCHASTICINTEGRAL The ﬁrst step of our construction is to identify the semi-martingale integral a(s)dW (s), for a adapted and W a semi-martingale, with a sum of (Sko- R K K rokhod) integrals of a and its stochastic gradient with respect to the driving Brownian motion B and time. 21

Proposition 6.1. Assume that a GIγ,θ,q,t and K is a semi-martingale ker- ∈ nel in Eγ,θ,p,t. Then 1,2 (i) The process It(a)( ) is in L (0,t) and · t Z It(a)(s)dBs It(a) L1,2(0,t) a γ,θ,q,t K γ,θ,p,t . 0 ≤ k k ≤ k k k k

~~(ii) We have the decomposition~~

~~t t t t (6.1) Z a(s)dWK(s) = Z Ita(s)dBs + Z Z Dsa(u)∂1K(u,s)duds. 0 0 0 s~~

(iii) If furthermore a CGIγ,θ,q,t then the process ∈ t s Jt(D (a)( ))(s) = Z Dsa(u)∂1K(u,s)du → · · s satisﬁes t t Z Jt(D (a)( ))(s) Z Jt(D (a)( ))(s) L2(Ω) ds 0 · · 2 ≤ 0 k · · k L (Ω)

~~C a γ θ K γ θ . ≤ |k k| , ,q,tk k , ,p,t and we have the upper bound~~

~~t (6.2) Z a(s)dWK(s) C a γ,θ,q,t K γ,θ,p,t . 0 2 ≤ |k k| k k L (Ω)~~

~~1,2 Proof. (i) On the one hand, Lemma 6.2 (ii) implies that Jt(a)( ) L (0,t) and · ∈~~

~~Jt(a) 1,2 a q 1,2 K γ θ . L (0,t) Lw γq (D ) , ,p,t k k ≤ k k − k k On the other hand, the second part of Proposition 2.9 implies that~~

K(t, )a( ) 2 2 Ω C a θ 2 Ω K γ θ . k · · kL ([0,T],L ( ,P)) ≤ k kL ∗ (L ( ))k k , ,p,T We conclude by combining these two results. (ii) Recall from Proposition 2.5 that WK is a semi-martingale with decomposition

~~dWK(s) = K(s,s)dBs +W∂1K(s)ds. Therefore, since a is adapted and in L1,2(0,t), we have that t t t~~

Z a(s)dWK(s) = Z a(s)K(s,s)dBs + Z a(s)W∂1K(s)ds 0 0 0 t t u = Z a(s)K(s,s)dBs + Z a(u)Z ∂1K(u,s)dBsdu. 0 0 0 22 P. CARMONA, L. COUTIN, AND G. MONTSENY We now apply the integration by parts formula of Malliavin Calculus (6.3) t t t u Z a(s)dWK(s) = Z a(s)K(s,s)dBs + Z Z a(u)∂1K(u,s)dBsdu 0 0 0 0 t u + Z Z Dsa(u)∂1K(u,s)dsdu. 0 0 def Since M2 = sup x ∂ K(x,u)2 du < +∞, we can apply Fubini’s Theo- x t R0 1 rem to the third term≤ on the right hand side of (6.3) to obtain t u t t Z Z Dsa(u)∂1K(u,s)dsdu = Z Z Dsa(u)∂1K(u,s)duds. 0 0 0 s Indeed, t u EZ Z Dsa(u) ∂1K(u,s) dsdu 0 0 | || | ≤ 1 t u 2 2 ME"Z Z Dsa(u) ds du# 0 0 | |

~~M a 1,2 . ≤ k kL The anticipating Fubini’s Stochastic Theorem (see Theorem 3.1 of Leon [17]) yields that t s t t Z Z a(u)∂1K(u,s)dBsdu = Z Z a(u)∂1K(u,s)dudBs . 0 0 0 s~~

~~Indeed, all we need to do is to consider the ﬁnite measure µ(dx) = dx1[0,t](x) on X = [0,t] and the measurable function~~

φ(x,u,ω) = a(x,ω)∂1K(x,u)(x,u t). ≤ It is clear that for fixed x [0,t], φ(x) : u φ(x,u,.) is in L1,2(0,t), with ∈ → Dvφ(x,u) = Dva(x)∂1K(x,u) and t t φ 2 E φ 2 φ 2 (x) L1,2 = Z (x,u) + Z (Dv (x,u)) dvdu k k 0 0 x x 2 2 2 = Z du∂1K(x,u) Ea(x) + Z (Dva(x)) dv. 0 0 Therefore the Bochner integral φ(x)dµ(x) is well defined since RX 2 2 Z µ(dx) φ(x) 1,2 µ(X)Z µ(dx) φ(x) 1,2 k kL ≤ k kL t x x 2 2 2 µ(X)EZ dx(a(x) + Z (Dva(x)) dv)Z ∂1K(x,u) du ≤ 0 0 0 x ∂ 2 ∞ µ(X)supZ 1K(x,u) du a L1,2(0,t) < + . ≤ x t 0 k k ≤ 23 (iii) The first part is a consequence of the first part of Proposition 2.9 applied to the processa ˜(u,s) = Dsa(u). We conclude by combining this upper bound with (i).

1,2 t Lemma 6.2. (i) For every a L (0,t) such that s a(v)∂1K(v,s)dv is ∈ → Rs in L1,2 we have the commutation relation t t Du Z a(v)∂1K(v,s)dv = Z Dua(v)∂1K(v,s)dv. s s K (ii) The application (a,K) Jt (a) is bilinear continuous from 1,2 q m,2 → m,2 L (0,t) Lw γq (D ) Eγ,θ,p,T to L with m = 0,1,2. ∩ − ×

~~Proof. (i) The simplest way to see this property is to use the Wiener chaos expansion of a~~

a(s) = ∑ Λm( fm(.,s)), m 0 ≥ where Λm denotes the m-th multiple Wiener integral and fm(s1,...,sm,s) are square integrable kernels symmetric in the ﬁrst m variables. By linearity, we can restrict ourselves to the case a = Λm( fm(.,s)) for some m 1. On the one hand, Fubini’s Stochastic Theorem implies that ≥ t t Z a(v)∂1K(v,s)dv = Z dv∂1K(v,s)Λm( fm( ,v)) s s • t = Λm Z ∂1K(v,s) fm( ,v)dv. • → s • Therefore, t t Du Z a(v)∂1K(v,s)dv = mΛm 1 Z ∂1K(v,s) fm( ,u,v)dv. s − • → s • On the other hand, t t Z Dua(v)∂1K(v,s)dv = Z ∂1K(v,s)mΛm 1( fm( ,u,v))dv s s − • t = mΛm 1 Z ∂1K(v,s) fm( ,u,v)dv. − • → s • (ii) For t [0,T], Proposition 2.9 implies ∈

(6.4) K K s J (a(.,s))(s) 2 + (r,s) J (Dra(.,s))(s) 2 2 → t L (Ω [0,T]) → t L (Ω [0,T] ) ≤ × × C a q 1,2 K γ θ . Lw γq (D ) , ,p,T k k − k k 24 P. CARMONA, L. COUTIN, AND G. MONTSENY We shall now use a by product of the proof of Proposition 2.5, namely the existence of a sequence of semi-martingale kernels (Λn)n N given by ∈ T n 1 γ Λn(r,s) = K(T,s) Z h (u s,s)(u + s)− du − r − n − N n φn ψn h (v,s) = ∑ i (v) i (s), i=0 φn ψn ∞ φn Λ where i , i are in L , the i i have disjoint support and limn ∞ K n γ,θ,p,T = 0. → → k − k It is clear (see (i)) that for every n, the random variable t Λn n 1 γ Jt (a( ,s))(s) = Z a(u,s)h (u s,s)(u + s)− du · s − n − belongs to D1,2 and that

Λn Λn (6.5) DrJ (a( ,s))(s) = J (Dra( ,s))(s). t · t · From the inequality (6.4) applied to the kernels Λn K,Λn Λm we de- Λn − − duce that the sequence (Jt (a( ,s))(s),s [0,T]) is a Cauchy sequence in 1,2 K· ∈ 2 Ω L (0,T) that converges to (Jt (a( ,s))(s),s [0,T]) in L ( [0,T]). · Λn ∈ × 2 Thanks to (6.5), we obtain also that (DrJt (a( ,s))(s),(s,r) [0,T] ) con- K 2 ·2 2 ∈ verges to (J (Dra(.,s))(s),(s,r) [0,T] ) in L (Ω [0,T] ). K ∈ 1,2 × Therefore, (Jt (a( ,s))(s),s [0,T]) is in L (0,T) and we have the commutation relation.· ∈ K K (6.6) DrJ (a( ,s))(s) = J (Dra( ,s))(s). t · t · It is now clear that (6.4) gives the proof of the second part of the Lemma for m = 1. For m = 2, we only need to replace a by Da.

~~Combining the density of Fγ,θ,p,t in Eγ,θ,p,t (see Proposition 2.5) with Propo- sition 6.1 yields that the continuous bilinear operator λt :CGIγ,θ,q,t Fγ,θ,p,t × → L2:~~

~~t λt(a,K) = Z a(s)dWK(s) 0 ˜ 2 can be uniquely extended to an operator λt : CGIγ,θ,q,t Eγ,θ,p,t L , de- ﬁned on the closure of this product space. × →~~

Deﬁnition 6.3. For a CGIγ,θ,q,t and K Eγ,θ,p,t the stochastic integral of ∈ ∈ a with respect to W is deﬁned to be λ˜ (a,K) and denoted by ta(s)dW (s). K t R0 K By construction we have the decomposition (6.1) and the upper bound (6.2). 25

~~7. THESTOCHASTICINTEGRALASAPROCESS We shall exhibit assumptions that ensure that the process t ta(s)dW (s) R0 K has a continuous modiﬁcation. →~~

β β 1 Theorem 7.1. Assume that for some > 0 such that + hγ > 2 : β K Aγ θ . • ∈ , ,p,T the adapted integrand a is in CβL1,2(0,T). • α sups T a L , for αhγ > 1. • ≤ | | ∈ t Then the process t 0Ita(s)dBs has a continuous modiﬁcation on [0,T]. → R t If, furthermore, α CGIγ θ then t a(s)dW (s) has a continuous , ,q,T R0 K modiﬁcation on [0,∈T]. →

Proof. Recall that the stochastic integral may be represented as t Z It(a)(s)dBs = X(t) +Y(t) 0 t X(t) = Z K(t,s)a(s)dBs 0 t 0 Y(t) = Z Jt(∆ a)(s)dBs . 0 The continuity of X is established via Kolmogorov’s continuity criterion. More precisely for α > 0 big enough, there exists η > 0 and C = C(T,α) such that α η (7.1) E X(t + τ) X(t) Cτ1+ (0 t < t + τ T). | − | ≤ ≤ ≤ First we write t t+τ X(t +τ) X(t) = Z (K(t +τ,s) K(t,s))a(s)dBs +Z K(t +τ,s)a(s)dBs − 0 − t Since a is adapted, we can apply Burkholder-Davis-Gundy inequalities to the martingales

~~r Z [K(t + τ,s) K(t,s)]1(0,t)(s)a(s)dBs → 0 −~~

r Z K(t + τ,s)1(t,t+τ)(s)a(s)dBs → 0 to obtain, for a constant Cα, the upper bound 1/α t α/2 α 1/α 2 2 E X(t + τ) X(t) CαE"Z [K(t + τ,s) K(t,s)] a(s) ds # | − | ≤ 0 − 1/α t+τ α/2 2 2 +CαE"Z K(t + τ,s) a(s) ds # t 26 P. CARMONA, L. COUTIN, AND G. MONTSENY The integrability assumptions on a, and Proposition 5.1 implies, for α > 2, τ τ X(t + ) X(t) Lα(Ω) Cα sups T a(s) Lα(Ω) K(t + , ) K(t, ) L2([0,T]) k − k ≤ k ≤ | |k k · − · k τhγ C sups T a(s) Lα(Ω) K γ,θ,p,T . ≤ k ≤ | |k k k Therefore we only need to assume also αhγ > 1 to obtain (7.1). The existence of a continuous modiﬁcation for the process Y is given by the Propo- sition 7.2.

~~If, furthermore, α CGIγ,θ,q,T , then the stochastic integral may be represented as ∈~~

~~t t t Z a(s)dWK(s) = Z It(a)(s)dBs + Z(t), Z(t) = Z Jt(D (a))(s)ds 0 0 0 · The process Z is obviously a continuous process so that ends the proof.~~

β β 1 ˜ β 1,2 Proposition 7.2. Let > 0 such that +hγ > 2 . Assume that a H (D ) β t K ∈ and K Aγ θ . Then the process t J (a( ,s))(s)dBs has a continu- ∈ , ,p,T → R0 t · ous modiﬁcation. β 1,2 ∆0 In particular, if a C L (0,T) then a˜(u,s) = s,ua = a(u) a(s) is in ˜ β 1,2 ∈ t K 0 − H (D ) and t J (∆ a)(s)dBs has a continuous modiﬁcation. → R0 t

~~K K Proof. According to Lemma 6.2 (ii) we have DrJ (a)(s) = J (Dra)(s) and~~

~~K K β+hγ β τ Jt+τ(a( ,s))(s) Jt (a( ,s))(s) L1,2 C a ˜ β(D1,2) K . · − · ≤ k kH k kAγ,θ,p,T~~

~~Since 2(β+hγ) > 1 we can conclude by Kolmogorov’s continuity criterion.~~

8. ITOˆ ’SFORMULA 1 Ito’sˆ formula for fractional Brownian motion of Hurst parameter H > 2 is now well known: see, for instance, Decreusefond and Ustunel [9], Dai and Heyde [8]. Here, we show how to obtain it for a wide family of kernels. The ﬁrst step of our method is to write Ito’sˆ formula for a semi-martingale kernel in a suitable way, that is a way that will be easily extended to more general kernels. 27

~~2 Proposition 8.1. Let K be a semi-martingale kernel and f Cb. Then, almost surely, for every t > 0: ∈ t (8.1) f (WK(t)) = f (0) + Z It( f 0(WK))(s)dBs 0 t 1 d 2 + Z f 00(WK(s)) E WK(s) ds. 2 0 ds~~

~~Proof. We let φ(s) = E W (s)2 = s K(s,r)2 dr . Since K is a semi-martingale K R0 kernel, φ is differentiable with~~

~~s 2 φ0(s) = K(s,s) + 2Z K(s,r)∂1K(s,r)dr . 0 By the chain rule,~~

~~Ds f 0(WK(u)) = f 00(WK(u))Ds(WK(u)) = f 00(WK(u))K(u,s),~~

~~1,2 and therefore the process f 0(WK(s)) is in L (0,t) and we may use Propo- sition 6.1. Furthermore,~~

~~t t t Z Jt(Ds f 0(WK))(s)ds = Z dsZ dv∂1K(v,s)Ds f 0(WK(v)) 0 0 s t t = Z dsZ dv∂1K(v,s) f 00(WK(v))K(v,s). 0 s~~

~~Recall that the semi-martingale decomposition of WK is~~

~~dWK(s) = K(s,s)dBs +W∂1K(s)ds.~~

~~Therefore, Ito’sˆ formula yields~~

t t 1 2 f (WK(t)) = f (0) + Z f 0(WK(s))dWK(s) + Z f 00(WK(s))K(s,s) ds 0 2 0 t t = f (0) + Z It( f 0(WK))(s)dBs + Z Jt(Ds f 0(WK))(s)ds 0 0 t 1 2 + Z f 00(WK(s))K(s,s) ds 2 0 t 1 t = f (0) + Z It( f 0(WK))(s)dBs + Z f 00(WK(s))φ0(s)ds. 0 2 0

~~β We shall make use of Bγ,θ,p,T deﬁned in 2.6. 28 P. CARMONA, L. COUTIN, AND G. MONTSENY~~

β β Theorem 8.2. Assume that for some < hγ the kernel K is in Bγ,θ,p,T for 4 1 every T > 0, that f Cb, and that hγ > 4 . Then, almost surely, for every t > 0, ∈ t (8.2) f (WK(t)) = f (0) + Z It( f 0(WK))(s)dBs 0 t 1 d 2 + Z f 00(WK(s)) E WK(s) ds. 2 0 ds

Proof. Our first step will be to prove that the processes on both sides have a continuous modification. Then, we shall establish this formula for fixed t > 0. Since WK has a continuous modification, see Proposition 5.1, f (WK) also has a continuous modification. It is obvious that t 1 d 2 t f (0) + Z f 00(WK(s)) E WK(s) ds, → 2 0 ds also has a continuous modification. In fact, Proposition 5.1 ensures that a.s. and in L2, sample paths of the Gaussian process WK are Holder¨ continuous with index β < hγ. Since f 0, f 00 (3) and f are bounded, Proposition 8.4 ensures that a = f 0(WK) belongs to β ∞ C L1,2(0,T) L ([0,T] Ω) as soon as β < hγ. We furthermore impose 1 ∩ × that β > hγ > γ 1/q, which is possible since hγ > 1/4; then, accord- 2 − − ing to Theorem 7.1, the process t tI ( f (W ))(s)dB has a continuous R0 t 0 K s modification. →

Let us now establish formula (8.2) for a ﬁxed t > 0. Assume that Kn is β a family of semi-martingale kernels such that Kn K in Bγ θ . Then → , ,p,T formula (8.2) is valid for Kn. We shall ﬁrst handle the term t d E 2 Z f 00(WKn (s)) WKn (s) ds. 0 ds According to the proof of Corollary 5.4, by taking a subsequence, we may 2 assume that sups t WKn (s) WK(s) 0 both in L and almost surely. There- fore according to≤ Lemma| 2.13:− | → t d t d E 2 E 2 Z f 00(WKn (s)) WKn (s) ds Z f 00(WK(s)) WK(s) ds. 0 ds → 0 ds To prove the convergence t t

Z It f 0(WKn )(s)dBs Z It f 0(WK(s))dBs 0 → 0 Kn 1,2 we need to prove that I an Ita in L (0,t) where a(s) = f (WK(s)) t → 0 and an(s) = f 0(WKn (s)). Thanks to Proposition 6.1, and to the fact that 29 β Kn K in Bγ θ , it is enough to prove that a an γ θ 0, and this → , ,p,T k − k , ,q,t n7−→∞ we achieve in the following Proposition 8.4. → Deﬁnition 8.3. From now on, ∆0F,∆1F will denote the Taylor expansion of the function F: 0 1 ∆ F = F(y) F(x), ∆ F = F(y) F(x) (y x)F0(x). x,y − x,y − − −

~~m+i+2 Proposition 8.4. Assume f Cb with i = 0,1 and m = 0,1,2. Then the i ∈ (i+1)hγ m,2 mapping K ∆ f is continuous from Eγ,θ,p,T to H (D ). → WK(s),WK(u)~~

Proof. Let X be a separable Hilbert space. Definition 8.5. A function G : R2 [0,T]2 X is said to satisfy assumption HH (m,k,k˜) if G is m-times differentiable× → with respect to its first two variables, and if there exists a constant C(G) = C(G,m,k,k˜) such that for every real numbers (x,y) R2, 0 u,s T, integers i, j such that 0 i + j m, we have ∈ ≤ ≤ ≤ ≤ j ∂i+ j max(k l,0) k˜ i, j G(x,y,u,s) C(G) ∑ y − u s . X ≤ = | | | − | l 0 M+m M 1 1 1 (M) Observe that if f C , then G(x,y,u,s) = ∆ − f = f (x + ∈ b x,x+y M! R0 θy)yM dθ satisfies HH (m,M,0). Therefore Proposition 8.4 is an immediate consequence of the following Lemma. Lemma 8.6. Assume α 1 and G satisfies HH (m,k;k˜). Then for every ≥ K Eγ,θ,p,T , 0 u,s T, ∈ ≤ ≤ m 0 k+ j khγ+k˜ G(WK(s),∆ WK,u,s) α C C(G) ∑ K γ θ u s . s,u Dn, (X) ≤ × × k k , ,p,T × | − | j=0 Proof. The proof is an induction on the integer m. Assume that G satisfies HH (0,k,k˜). Then, 0 0 k k˜ G(WK(s);∆ WK;u,s) C(G) ∆ WK u s . s,u X ≤ s,u | − | 0 0 Since the random variable ∆ WK is centered with covariance ∆ K( ,r) s,u s,u · L2([0,T],dr) with the convention K(u,r) = 0 for u < r, we deduce from Proposition 2.10 the upper bound 0 hγ (8.3) ∆ K( ,r) C K γ θ u s . s,u · L2([0,T],dr) ≤ k k , ,p,T | − |

~~We plug in the expression of the kα-th order moment of a centered Gaussian random variable, to obtain that the Lemma is satisﬁed for a constant k C0 = C N (0,1) . khγ~~

30 P. CARMONA, L. COUTIN, AND G. MONTSENY We now assume the Lemma satisfied for m and let G verify assumption HH (m + 1,k,k˜). Then G is differentiable with respect to the first two variables up to the order m + 1, and G together with its partial derivatives have at most polynomial growth. According to the chain rules of Malliavin Cal- ∆0 m+1,α culus, the random variable G(WK(s), s,uWK;u,s) belongs to D and m+1 0 i, j,σ 0 D G(WK(s),∆ WK;u,s) = ∑ G (WK(s),∆ WK;u,s), s,u s,u σ Sm+1,i+ j=m+1 ∈ where Sn denotes the group of order n permutations, the tensor product and ⊗ (8.4) 1 i+ j i G(i, j,σ)(x,y,u,s)(r1,...,rm+1) = ∂ G(x,y,u,s) K(s, )(rσ ,...,rσ ) i! j! i, j ⊗ · (1) (i) j 0 ∆ K( , )(rσ ,...,rσ ). ⊗ s,u · · (i+1) (m+1) Injecting the upper bound 8.3, we get that G(i, j,σ) satisfies HH (0,k m+1 − j,k˜ + jhγ) on X = L2([0,T] ). We can therefore apply the Lemma for m = 0 to every G(i, j,σ) to obtain

m+1 0 (8.5) D G(WK(s),∆ WK;u,s) α s,u L (X) ≤ k+ j khγ+k˜ CC(G,m + 1)∑ K γ,θ,p,T u s . j k k | − | We conclude by using the deﬁnition of the norm in Dα,m+1 and the induction assumption. Lemma 8.7. Assume that G satisﬁes HH (m,k,k˜) with m 1,k > 0. Then 0 ≥ ˜ khγ+k˜ m 1,2 the mapping K G(WK(s),∆ WK,u,s) is continuous from Eγ,θ,p,T to H (L ) → s,u − (with the convention L0,2 = L2(Ω)).

Proof. Using Taylor integral expansion we have for K,K Eγ,θ,p,T 0 ∈ 1 ∆0 ∆0 θ (8.6) G(WK(s); s,uWK;u;s) G(WK (s); s,uWK ;u;s) = Z Bθ d , − 0 0 0 with ∇ ∆0 ∆0 t Bθ = G(WK+θ(K K )(s); s,uWK+θ(K K );u,s) WK K (s); u,sWK K − 0 − 0 − 0 − 0 where xt denotes the transpose of x. For i = 1,2, the mapping ∂iG(x,y,u,s) satisﬁes HH (m 1,k + 1 i,k˜). Therefore Lemma 8.6 implies − −

∂ ∆0 (8.7) iG(WK+θ(K K ); s,uWK+θ(K K );s,u) m 1,2 − 0 − 0 L − m 1 (k+1 i)hγ+k˜ − k+ j k+ j C u s − ∑ ( K γ θ + K0 ). ≤ | − | k k , ,p,T γ,θ,p,T j=0 The same Lemma 8.6 yields 31

~~(8.8) WK K (s) Lm 1,2 C K K0 γ θ k − 0 k − ≤ − , ,p,T~~

~~∆0 W C K K u s hγ . s,u K K0 m 1,2 0 γ,θ,p,T − L − ≤ − | − | Combining (8.7) and (8.8) yields,~~

~~m 1 − k+ j k+ j khγ+k˜ Bθ m 1,2 C ∑ ( K γ θ + K0 γ θ ) K K0 u s k kL − ≤ k k , ,p,T , ,p,T − | − | j=1 and injecting this into (8.6) yields the desired result.~~

8.1. Application to fractional Brownian motion. According to Lemma 2.7 ε ε 1 we can take hγ = H with > 0 small enough so that hγ > 4 and then use Theorem 8.2. − 1 8.2. Ito formula for H > 6 . We ﬁrst derive a new expression for the Ito formula for the Gaussian processes associated to semi-martingale kernels. 4 Proposition 8.8. Let K a semi-martingale kernel and f Cb. Then, for t [0,T] almost everywhere, ∈ ∈ 1 t d f (WK(t)) = f (0) + Z f 00(WK(s)) RK(s,s)ds 2 0 ds t + Z K(t,s) f 0(WK(s)) + f 00(WK(s))E[WK(t) WK(s) Fs] dBs 0 − | t Z K(t,s) f 000(WK(s))E[(WK(t) WK(s))WK(s)]dBs − 0 − t K 1 + I ∆ f 0 + f 000(W (s))E[(W ( ) W (s))W (s)] (s)dB Z t h WK(s),WK( ) K K K K i s 0 · · − t s t ∂ Z Z f 00(WK(s)) f 00(WK(s1)) Z 1K(u,s1)K(u,s)dudBs1 dBs − 0 0 − s with RK the covariance function of the Gaussian process WK

RK(x,y) = E[WK(x)WK(y)], 4 Proof. Let K a semi-martingale kernel and f Cb. According to Proposi- tion 8.1, for t [0,T], we have ∈ ∈ t K f (WK(t)) = f (0) + Z It ( f 0(WK))(s)dB(s) 0 t 1 d 2 + Z f 00(WK(s)) E WK(s) ds, 2 0 ds A Taylor expansion of f 0 between WK(u) and WK(s) yields IK( f (W ))(s) = K(t,s) f (W (s)) + IK(∆1 f )(s) t 0 K 0 K t WK(s),WK( ) 0 · K ∆0 + f 00(WK(s))It ( s, WK)(s). · 32 P. CARMONA, L. COUTIN, AND G. MONTSENY

~~4 2,2 Since f Cb, the process (a(s) = f 00(WK(s)),0 s T) belongs to L and ∈ ≤ ≤~~

~~s t t d ∂ E Z Ds1 a(s)Z K(u,s) 1K(u,s1)duds1 = f 000(WK(s))Z K(u,s) [WK(u)WK(s)]du. 0 s s ds Using the integration by parts formula,~~

~~s t ∂ Z Ds1 a(s)Z K(u,s) 1K(u,s1)duds1 0 s 0 = f 000(WK(s)) K(t,s)E ∆ WK WK(s) s,t K ∆0 Jt (E s, WKWK(s) )(s) . − · Then Lemma 8.9 yields,~~

1 t d f (WK(t)) = f (0) + Z f 00(WK(s)) RK(s,s)ds 2 0 ds t ∆0 + Z f 0(WK(s)) + f 00(WK(s))E s,tWK Fs K(t,s)dBs 0 h | i t ∆0 + Z f 000(WK(s))E s,tWK WK(s) K(t,s)dBs 0 h i t K 1 0 + I ∆ f 0 + f 000(W (s))E ∆ W W (s) (s)dB Z t h WK(s),WK( ) K s, K K i s 0 · · t s ∆0 K + Z Z W (s),W (s ) f 00 Jt (K( ,s))(s1)dBs1 dBs 0 0 K K 1 ·

Lemma 8.9. Let K be a semi-martingale kernel and a L2,2 an adapted process. Then ∈ t t K ∆0 ∆0 Z a(s)It ( s, WK)(s)dBs = Z a(s)E s,tWK Fs K(t,s)dBs 0 · 0 | t s t ∆0 ∂ + Z Z s,s a Z K(u,s) 1K(u,s1)dudBs1 dBs 0 0 1 s t s t ∂ Z Z Ds1 a(s)Z K(u,s) 1K(u,s1)duds1dBs . − 0 0 0

0 Proof. We split WK(u) WK(s) = ∆ WK in two parts: − s,u 0 0 0 0 ∆ WK = E ∆ WK Fs + ∆ WK E ∆ WK Fs s,u s,u | s,u − s,u | E ∆0 Observe that WKE = ( s,tWK Fs ,s t T) is a centered Gaussian process that can be written, thanks| to Fubini’s ≤ ≤ Stochastic Theorem, 33

~~s ∆0 WKE (t) = Z s,tK( ,s1)dBs1 0 · t u s ∧ ∂ = Z duZ 1K(u,s1)dBs1 s 0~~

~~The integration by parts formula (applied to the semi-martingale WKE and the deterministic kernel K( ,s)) yields · t t ∂ K(t,s)WKE (t) = K(s,s)WKE (s)+Z WKE (u) 1K(u,s)du+Z K(u,s)dWKE (u) s s~~

~~Since a is adapted and WKE (s) = 0, we obtain~~

t ∆0 ∂ Z a(s)E s,uWK Fs 1K(u,s)du s | t E ∆0 = a(s) s,tWK Fs K(t,s) Z a(s)1(s,t)(u)K(u,s)dWKE (u) | − 0 0 = a(s)E ∆ WK Fs K(t,s) s,t | t s ∂ Z a(s)K(u,s)Z 1K(u,s1)dBs1 du. − s 0 The Integration by parts formula yields then, as in the proof of Proposi- tion 6.1

t ∆0 ∂ Z a(s)E s,uWK Fs 1K(u,s)du = a(s)E[WK(t) WK(s) Fs]K(t,s) s | − | s t ∂ Z a(s)Z K(u,s) 1K(u,s1)dudBs1 − 0 s s t ∂ Z Z Ds1 a(s)K(u,s) 1K(u,s1)duds1 . − 0 s Integrating with respect to dB(s) each term of this equality we obtain, t t (8.9) Z a(s)Z E[WK(u) WK(s) Fs]∂1K(u,s)dudBs = 0 s − | t Z a(s)E[WK(t) WK(s) Fs]K(t,s)dBs 0 − | t s t ∂ Z Z a(s)Z K(u,s) 1K(u,s1)dudBs1 dBs − 0 0 s t s t ∂ Z Z Z Ds1 a(s)K(u,s) 1K(u,s1)duds1dBs − 0 0 s Observe that for s t, ≤ t WK(u) WK(s) E[WK(u) WK(s) Fs] = Z K(u,s1)dB(s1) − − − | s 34 P. CARMONA, L. COUTIN, AND G. MONTSENY and using again stochastic integration by parts and Fubini’s Stochastic The- orem for deterministic integrands, we obtain that t t Z Z a(s)(WK(u) WK(s) E[WK(u) WK(s) Fs])∂1K(u,s)dudBs = 0 s − − − | t s t 1 ∂ Z Z a(s)Z K(u,s1) 1K(u,s)dudBsdBs1 . 0 0 s

Summing up this equality with (8.9) yields, after one substitution s s1, ↔ t K Z a(s)It (WK( ) WK(s))(s)dBs = a(s)E[WK(t) WK(s) Fs]K(t,s) 0 · − − | t s t ∂ Z Z (a(s) a(s1))Z K(u,s) 1K(u,s1)dudBs1 dBs − 0 0 − s t s t ∂ Z Z Ds1 a(s)Z K(u,s) 1K(u,s1)duds1dBs . − 0 0 0

~~β 1 Proposition 8.10. Assume + 2hγ > 2 . Then the mapping t K ∂ (K,K0,a) a(s,s1)Jt (K0( ,s))(s1)1(s1~~

~~t s K Z dB(s)Z a(s,s1)Jt (K0( ,s))(s1) dB(s1) 0 0 · 2 ≤ L β C a ˜ β K K0 γ,θ,p,T . k kH k kAγ,θ,p,T~~

~~Moreover, the process~~

~~t s t ∂ (8.10) t Z dB(s)Z a(s,s1)Z K0(u,s) 1K(u,s)dudBs1 → 0 0 s has a continuous modiﬁcation on [0,T].~~

~~Proof. According to the deﬁnition (2.17) of J2,t we have~~

~~K K,K0 (8.11) a(s,s1)J (K0( ,s))(s1)1 = J (a)(s,s1). t · (s1~~

~~K0,K J (a)(s,s ) C a ˜ β K β K0 . 2,t 1 2,2 γ,θ,p,T L ≤ k kH k kAγ,θ,p,T~~

~~Moreover, thanks to the same Lemma 2.14, if 35~~

~~t s K X(t) = Z dB(s)Z a(s,s1)J (K0( ,s))(s)1(s1~~

~~β+2hγ τ β τ X(t + ) X(t) L2 C a ˜ β K K0 γ,θ,p,T k − k ≤ k kH k kAγ,θ,p,T~~

~~Since 2(β + 2hγ) > 1, Kolmogorov’s continuity criterion ensures the existence of a continuous modiﬁcation.~~

~~We can now state and prove an Ito Formula that applies to the family of 1 Gaussian processes WK such that hγ > 6 and therefore, by Lemma 2.7 to 1 fractional Brownian motion with Hurst index H > 6 .~~

β 1 6 Proposition 8.11. Let K Bγ θ with β/2 > hγ > and let f C . Then ∈ , ,p,T 6 ∈ b almost surely, for 0 < t T, ≤t 1 d 2 f (WK(t)) = f (0) + Z f 00(WK(s)) E WK(s) ds 2 0 ds t + Z f 0(WK(s)) + f 00(WK(s)E[WK(t) WK(s) Fs] K(t,s)dB(s) 0 − | t Z f 000(WK(s))E[(WK(t) WK(s))WK(s)] K(t,s)dB(s) − 0 − t K 1 + I ∆ f 0 + f 000(W (s))E[(W ( ) W (s))W (s)] (s)dB Z t WK(s),WK( ) K K K K s 0 · · − t s ∆0 K + Z dBs Z s,s f 00(WK)It (K( ,s))(s1)dBs1 0 0 1 ·

Proof. The first step of the proof consists in showing that every process involved in this formula has a continuous modification. Then we shall prove in a second step that the formula is valid for each fixed t by establishing the continuity with respect to K Fγ,θ,p,T for the norm γ,θ,p,T and using ∈ k·k β Proposition 8.8. We conclude by using the density of Fγ,θ,p,T into Bγ,θ,p,T .

First Step : existence of a continuous modification. 1. Since f (4) is bounded, Proposition 8.4 entails that ∆1 f be- WK(s),WK(u) 0 longs to H˜ 2hγ (D1,2). Hence, we infer from Proposition 7.2 that since 1 3hγ > 2 , the process t t IK(∆1 f )(s)dB Z t WK(s),WK( ) 0 s → 0 · has a continuous modification. 36 P. CARMONA, L. COUTIN, AND G. MONTSENY 2. Similarly, since f C4, Lemma 2.15 implies that we may apply Propo- ∈ b sition 7.2 to the process ( f 000(WK(s))E[(WK(u) WK(s))WK(s)]; 0 u,s T) to obtain the existence of a continuous− modification for ≤ ≤ t K t Z It ( f 000(WK(s))E[(WK(u) WK(s))WK(s)])(s)dBs . → 0 − 5 0 3. Since f C we can apply Proposition 8.10 to ∆ f 00 to ob- ∈ b WK(s),WK(s1) tain the existence of a continuous modification for t s ∆0 K t Z dBs Z s,s f 00(WK)It (K( ,s))(s1)dBs1 . → 0 0 1 · 4. The existence of a continuous modification for t t Z f 00(WK(s))E[WK(t) WK(s) Fs]K(t,s)dBs → 0 − | is a consequence of Burkholder-Davis-Gundy inequalities, Proposi- tion 5.1, and the fact that f 00 is bounded. 3 5. Eventually, since f Cb the existence of a continuous modification for ∈ t t Z f 000(WK(s))E[WK(s)(WK(t) WK(s))]K(t,s)dBs → 0 − is a consequence of Burkholder-Davis-Gundy inequalities and Lemma 2.15. Second Step : Ito’sˆ formula for a fixed time. According to the defini- β t d E 2 tion 2.6, page 8, of Bγ,θ,p,T , the mapping K 0 f 00(WK(s)) ds WK(s) ds β → R is continuous on Bγ,θ,p,T . In order to establish the continuity of t K Z f 0(WK(s)) + f 00(WK(s)E[WK(t) WK(s) Fs] K(t,s)dB(s) → 0 − | t Z f 000(WK(s))E[(WK(t) WK(s))WK(s)] K(t,s)dB(s) − 0 − α we only need to prove the continuity from Eγ,θ,p,T into the space L , for some α θ /2, of the integrand mapping ≥ ∗ K f 0(WK(s))+ f 00(WK(s)E[WK(t) WK(s) Fs] f 000(WK(s))E[(WK(t) WK(s))WK(s)]. → − | − − This can be obtained by Taylor expansions for f 0, f 00 and f 000, Proposi- tion 5.1, Lemma 2.15 and the following upper bound, proved for any adapted θ /2 process b bounded in L ∗ via Burkholder-Davis-Gundy and Holder¨ (θ/2,θ∗/2) inequalities θ t T 1/ ∗ θ∗/2 Z K(t,s)b(s)dBs C K γ,θ,p,T Z E b(s) ds . 0 2 ≤ k k 0 h| | i L

Thanks to Lemma 6.2, the continuity of t K IK(∆1 f )(s)dB Z t WK(s),WK( ) 0 s → 0 · 37 is a consequence of the continuity of 1 ˜ β 2,2 K Eγ,θ,p,T ∆ f 0 H (D ) ∈ → WK(s),WK(u) ∈ 1 for some β > hγ . This latter continuity is an easy consequence of Propo- − q sition 8.4 applied to β = 2hγ. Since hγ > 1/6, combining Lemmas 6.2 and 2.15 with a Taylor expansion of f 000, shows that the mapping t K K Z It ( f 000(WK(s))E[(WK( ) WK(s))WK(s)])(s)dBs → 0 · − hγ 2 Ω P is continuous from Aγ,θ,p,T into L ( , ). Eventually, thanks to Proposition 8.4, we can apply Proposition 8.10 to a(s,s1) = f (WK(s)) f (WK(s1)) to get the continuity of 0 − 0 t s ∆0 K K Z dBs Z s,s f 00(WK)It (K( ,s))(s1)dBs1 → 0 0 1 · 2hγ 2 Ω P from Aγ,θ,p,T into L ( , ).

APPENDIX A. PATHWISE VERSUS STOCHASTIC INTEGRATION H H We shall establish that for W1 , W2 two independent fractional Brownian 1 motions of index 0 < H < 2 , t H H Z W1 (s)dW2 (s) 0 cannot be deﬁned as a classical pathwise integral. To this end we recall that if Cα[0,T] denotes the space of Holder¨ continuous functions on [0,T] of index α then: α (A.1) for every α (0,H), a.s W H C [0,T]. ∈ ∈ (A.2) P W H CH[0,T] = 0. ∈ ((A.1) can be proved with Kolmogorov’s continuity criterion, and (A.2) is an immediate consequence of the law of the iterated logarithm, Theorem 1.7 of [2]). Let v( f ) be the variation index of the function f :

v( f ) = inf p > 0 : vp( f ) < +∞ . Then, see Theorem 5.3 of [11], 1 (A.3) almost surely v(W H) = . H t H H (1) 0W1 (s)dW2 (s) cannot be deﬁned as a p-variation integral. Indeed, R 1 1 see [11], this requires that for p,q > 0 such that p + q > 1, H H W Wp([0,T]), W Wq([0,T]). 1 ∈ 2 ∈ From (A.3) we deduce that p,q 1 and thus H > 1 . ≥ H 2 38 P. CARMONA, L. COUTIN, AND G. MONTSENY (2) tW H(s)dW H(s) cannot be deﬁned as a generalized stochastic inte- R0 1 2 gral of Russo-Vallois [28] and Zahle¨ [34]. Indeed, this integral requires H that the integrand W1 admits a generalized quadratic variation, and this im- 1 H 2 plies by Proposition 5.1 [34], that W1 is in the function space W2,∞−. Since, see Theorem 1.1 [34], for ε > 0 small enough

1 ε 1 1 2 2ε 3ε W ∞− I 2 − (L2) C 2 − , 2, ⊂ ⊂ H 1 3ε Hence, we need that W C 2 for arbitrary small ε, and this implies 1 ∈ − H 1 . ≥ 2 (3) tW H(s)dW H(s) cannot be deﬁned as a generalized integral of Ciesiel- R0 1 2 ski and al [4]. If this were the case, then section V.B implies that

~~H 1 H 1 1 1 W1 Bp,−1 for < H < = 1 . ∈ p p0 − p 1 H 1 H ε 1 H ε Since, for ε > 0 small enough, B − B∞ ∞ = C , this implies p,1 ⊂ −, − − − again H 1 . ≥ 2~~

APPENDIX B. AN ELEMENTARY CALCULUS LEMMA This Lemma is used in section 2 to evaluate the Holder¨ exponent of some integral processes. Lemma B.1. Let, for 0 s t T and γ (0,1) (1,2), ≤ ≤ ≤ ∈ ∪ γ f (s,t) = Z dudv(v u)− . 0

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~~PHILIPPE CARMONA,LABORATOIRE DE STATISTIQUE ET PROBABILITES´ ,UNIVER- SITE´ PAUL SABATIER, 118 ROUTEDE NARBONNE, 31062 TOULOUSE CEDEX 4 E-mail address: [email protected]~~

~~LAURE COUTIN,LABORATOIRE DE STATISTIQUE ET PROBABILITES´ ,UNIVERSITE´ PAUL SABATIER, 118 ROUTEDE NARBONNE, 31062 TOULOUSE CEDEX 4 E-mail address: [email protected]~~

~~GERARD´ MONTSENY, L.A.A.S,7, AVENUEDU COLONEL ROCHE, F-31077 TOULOUSE CEDEX 4 E-mail address: [email protected]~~